Image Processing Reference
In-Depth Information
S
OLUTION
The characteristic polynomial of A is
l
0
:
75
0
:
50
:
75
3
2
P(
l
)
¼jlI Aj¼
0
:
5
l
33
¼ l
1
:
75
l
þ
0
:
875
l
0
:
125
0
:
5
2
l þ
2
¼
(
l
1)(
l
0
:
5)(
l
0
:
25)
The eigenvalues are
l
1
¼
1,
l
2
¼
0
:
5,
and
l
3
¼
0
:
25. The corresponding
eigenvectors are
2
4
3
5
,
2
4
3
5
,
2
4
3
5
0
3
2
1
1
1
1
2
2
x
1
¼
x
2
¼
and
x
3
¼
Therefore,
2
4
3
5
2
4
3
5
2
4
3
5
1
011
(1)
k
011
0
0
)M
1
5)
k
f (A)
¼ Mf (
L
¼
312
212
0(0
:
0
312
212
25)
k
0
0
(0
:
2
4
3
5
2
4
3
5
5)
k
25)
k
01
1
0(0
:
(0
:
5)
k
55)
k
¼
22
3
3(0
:
2(0
:
5)
k
2(0
25)
k
1
23
2(0
:
:
2
4
3
5
5)
k
25)
k
5)
k
25)
k
5)
k
25)
k
2(0
:
(0
:
2(0
:
2(0
:
3(0
:
þ
3(0
:
5)
k
25)
k
3
5)
k
25)
k
5)
k
25)
k
¼
2(0
:
2(0
:
þ
2(0
:
4(0
:
3
3(0
:
þ
6(0
:
5)
k
25)
k
2
5)
k
25)
k
5)
k
25)
k
2(0
:
2(0
:
þ
2(0
:
4(0
:
2
3(0
:
þ
6(0
:
Hence,
2
4
3
5
5)
k
25)
k
5)
k
25)
k
5)
k
25)
k
2(0
:
(0
:
2(0
:
2(0
:
3(0
:
þ
3(0
:
A
k
5)
k
25)
k
5)
k
25)
k
5)
k
25)
k
¼
2(0
:
2(0
:
3
þ
2(0
:
4(0
:
3
3(0
:
þ
6(0
:
5)
k
25)
k
2
5)
k
25)
k
5)
k
25)
k
2(0
:
2(0
:
þ
2(0
:
4(0
:
2
3(0
:
þ
6(0
:
3.11.5 M
ATRIX
E
XPONENTIAL
F
UNCTION
e
At
The matrix exponential function e
At
is a very important function with applications in
analysis and design of continuous time-control systems. It is de
ned as an in
nite
series:
A
2
t
2
2
A
3
t
3
3
A
n
t
n
n
!
þ
e
At
¼
I
þ
At
þ
!
þ
!
þþ
(
3
:
192
)
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